On the Littlewood Problem Modulo a Prime

Green, Ben ; Konyagin, Sergei

Canadian journal of mathematics, Tome 61 (2009) no. 1, pp. 141-164

Résumé

Let $p$ be a prime, and let $f:\mathbb{Z}/p\mathbb{Z}\to \mathbb{R}$ be a function with $\mathbb{E}f=0$ and $||\hat{f}|{{|}_{1}}\le 1$ . Then ${{\min }_{x\in \mathbb{Z}/p\mathbb{Z}}}|f\left( x \right)|=O{{\left( \log p \right)}^{-1/3+\in }}$ . One should think of $f$ as being “approximately continuous”; our result is then an “approximate intermediate value theorem”.As an immediate consequence we show that if $A\subseteq \mathbb{Z}/p\mathbb{Z}$ is a set of cardinality $\left\lfloor {p}/{2}\; \right\rfloor $ , then ${{\sum }_{r}}\widehat{|\,{{1}_{A}}}\left( r \right)|\gg {{\left( \log p \right)}^{1/3-\in }}$ . This gives a result on a “ $\,\bmod \,p$ ” analogue of Littlewood's well-known problem concerning the smallest possible ${{L}^{1}}$ -norm of the Fourier transform of a set of $n$ integers.Another application is to answer a question of Gowers. If $A\,\subseteq \,{\mathbb{Z}}/{p\mathbb{Z}}\;$ is a set of size $\left\lfloor {p}/{2}\; \right\rfloor $ , then there is some $x\,\in \,\mathbb{Z}/p\mathbb{Z}$ such that $$||A\cap \left( A+x \right)\,-\,p/4|\,=o\left( p \right).$$

Détail

DOI : 10.4153/CJM-2009-007-4

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